I'll do a bit more of my own research into the spacesuit systems. Thanks for the guidance.

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As to the algae, my wife has 5+ fish tanks and is looking to add more. I'm not familiar with the exact Genus.Species of the algae that grows on her tanks. The more interesting scenario is how a tank of water (with or without fish) would work in 0g. The tank couldn't be "full" of water since the water'd float unless 100% contained. However, if contained, there'd also be no air transfer. Additionally, with no gravity to pull the water down and allow the air to rise, any gasses in the water would simply bubble around the tank rather than coalesce at the top. A fish tank for air is more of a colony solution in a gravity well.

Fish stores do sell "algae disks" for algae eaters such as plecostomuses (plecos) which are quite efficient and thorough at cleaning the tank. We have one which we transfer from tank to tank as he cleans it. Still, there's not enough algae for him. Hence the discs.

An idea that I saw one time using fish in space for a hydroponic type of arrangement kept the entire tank pressurized and flowing. The goldfish were in a sealed tank with a small inlet and outlet. Water was pumped into the inlet. The algae ridden water was then pumped out the outlet where it went to a high-pressure mister for the plant roots. The extra water then cycled back around to feed the inlet chamber of the tank.

1) So the algae tank needs to be a misted tank with corrugated sides and baffles. The mist keeps a high humidity (think rainforest) to allow the algae to grow while the baffles and sides provide a place for the algae to grow on. The open cylinder would also allow air to flow through it. If Chlorella only grows "dispersed in solution", then it'd be 'no-go' for the above reasons. Then we'd need to look at the best algae that grows in high humidity anchored on a substrate.

2) Another option may be to have a small opening with a semi-permeable layer such as a sponge which will keep the water contained in the tank but allow some air passage.

3) Since light and moisture are both parts of the equation, perhaps a clear, spiral, plastic, large-diameter tube could be used.
* Clear would allow for light penetration (real or artificial).
* Spiral would allow more tubing to be placed in a smaller area of the craft.
* Plastic is cheap and water/air proof.
* The larger the diameter of the tube, the more air could flow through it. The smaller the diameter of the tube, the better the processing of each cubic inch of air. A balance needs to be found.
From this core beginning, we'd have a mister at the beginning of the tube which mixes with a fan/air pump for cabin air. The air'd flow down the tube(s) then be collected at the other end.
A) Here, we could separated the excess humidity from the air. The air is dispersed back into the cabin while the water is recycled back to the mister.
B) The other option is to have the moisture come from a dehumidifier (installed elsewhere in the cabin) which separates the water from air for us. The mister wouldn't need to be in continual operation so the gradual loss of water in the tank could be compensated for by the dehumidifier. The cabin'll need some humidty anyway but not excessive.

The bigger question is going to be:
How much algae is necessary to process the CO2 output for one person in one hour? From there, you have the necessary parameters to be able to build the air scrubber. If we go with the tube option, then we'll have a relatively strong beginning for an estimate of tube length per person, number of tubes for the crew size, and the volume of air flow required through the tubes.

Much of this hardware might actually be already commercially available. It's simply not put together the way we're planning to use it.

I need to read through your suggestions more slowly. Obviously, some algae do grow attached to substrates, and a water spray would be a useful way to get water, and nutrients to them. These could also be supplied through porous support structures. I mention Chlorella, because it has long been used in the highest photosynthetic efficiency experiments. As this is an unattached, unicellular species, simple dispersion in agitated nutrient solution seems to be the preferred arrangement. These solutions are filtered or centrifuged to harvest Chlorella for human or animal food. The concentration range seems to be limited by the system, including the need for light to penetrate the growth volume. Thus thin tubes or growth channels permit much higher concentration. Illumination from both sides would improve illumination uniformity and increase the limiting concentration or channel thickness.

In addition, a very real photosynthetic saturation is mentioned, so that efficiency is improved with light levels considerably below that of direct sunlight. I doubt that there is an illumination TIME saturation, so 24 hour illumination should double or triple the yield. Even without this, growth quadrupling the mass in less than 24 hours is mentioned. The photosynthetic efficiency is so high (20% reported, with 39% actual conversion of monochromatic light energy to food energy content (Calories)) that a 2 foot by 2 foot surface in near Earth space could be sufficient to supply one average man. With doubling of the algae mass in less than twelve hours, this food source would involve less than one pound of dried Chlorella dispersed in the nutrient solution. This probably needs less than 20 pounds of solution, with shallow channels. It is feasible to concentrate sunlight with a lens into a cylindrical light guide with one hundred times smaller diameter than the collecting lens. Fresnel collecting lenses can be very light, and the thin light guides even lighter. Reflector terminated light guides can be arranged to scatter light quite uniformly along their length into the adjacent volume, permitting excellent illumination of the growth solution.

Enough of the optical ramblings: accept for now the fact that practical optical arrangements exist and can be operated in “zero g”. My first concern is whether a robust growth mass of Chlorella can be sustained without severe difficulties, and that the observed photosynthetic efficiency is within an order of magnitude of the theoretical projections. An eight foot square “interplanetary farm”, supporting a near Earth asteroid “prospector”, would be good enough! The preferred laboratory system for growing Chlorella seems to be to bubble CO2 enriched air through glass or plastic tubes of nutrient, with the bubbling action providing the desired agitation. The Chlorella could be removed from solution samples with filter paper, and dried and weighed to document the growth of biomass. More complex systems can wait for this demonstration of useful growth.